Question

solve for b.

(image of triangle with vertices b, a, c. side a = 18 (bc), angle at b is 28°, angle at a is 83°, side b is ac. law of sines: \\(\frac{\sin a}{a} = \frac{\sin b}{b} = \frac{\sin c}{c}\\)
round your answer to the nearest hundredth.

b = ?

Explanation:

Step1: Find angle C

The sum of angles in a triangle is \(180^\circ\). So, \(\angle C = 180^\circ - 28^\circ - 83^\circ = 69^\circ\). Wait, no, wait. Wait, side \(a\) is opposite angle \(A\)? Wait, no, in the Law of Sines, \(\frac{\sin A}{a}=\frac{\sin B}{b}\). Wait, let's check the labels. In triangle \(ABC\), angle \(B\) is \(28^\circ\), angle \(A\) is \(83^\circ\), side \(a\) is \(BC\) (opposite angle \(A\)), side \(b\) is \(AC\) (opposite angle \(B\))? Wait, no, the standard notation is side \(a\) opposite angle \(A\), side \(b\) opposite angle \(B\), side \(c\) opposite angle \(C\). Wait, in the diagram, side \(a = 18\) is between \(B\) and \(C\)? Wait, no, the label: \(a = 18\) is \(BC\), angle \(B\) is \(28^\circ\), angle \(A\) is \(83^\circ\). So angle \(A\) is at vertex \(A\), angle \(B\) at \(B\), angle \(C\) at \(C\). So side \(a\) is opposite angle \(A\) (so \(a = BC\)), side \(b\) is opposite angle \(B\) (so \(b = AC\)), side \(c\) is opposite angle \(C\) (so \(c = AB\)). Wait, but the Law of Sines is \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). Wait, let's confirm the angles. Angle \(A = 83^\circ\), angle \(B = 28^\circ\), so angle \(C = 180 - 83 - 28 = 69^\circ\). But side \(a = 18\) is opposite angle \(A\) (angle \(A = 83^\circ\)), so \(a = 18\) is opposite \(83^\circ\), and side \(b\) is opposite angle \(B = 28^\circ\)? Wait, no, that can't be, because then \(b\) would be smaller than \(a\), but let's check the formula. Wait, the Law of Sines: \(\frac{\sin A}{a}=\frac{\sin B}{b}\). So \(A = 83^\circ\), \(a = 18\), \(B = 28^\circ\), solve for \(b\). So rearranging, \(b=\frac{a\sin B}{\sin A}\). Let's compute that.

Step2: Apply Law of Sines

Given \(a = 18\), \(\angle A = 83^\circ\), \(\angle B = 28^\circ\). Using \(\frac{\sin A}{a}=\frac{\sin B}{b}\), we can solve for \(b\):

\(b=\frac{a\sin B}{\sin A}\)

Substitute the values: \(a = 18\), \(\sin B=\sin 28^\circ\), \(\sin A=\sin 83^\circ\)

Calculate \(\sin 28^\circ \approx 0.4695\), \(\sin 83^\circ \approx 0.9925\)

Then \(b=\frac{18 \times 0.4695}{0.9925}\)

First, multiply numerator: \(18 \times 0.4695 = 8.451\)

Then divide by \(0.9925\): \(8.451 \div 0.9925 \approx 8.51\) (rounded to nearest hundredth)

Wait, but wait, maybe I mixed up the sides. Wait, maybe side \(a\) is opposite angle \(B\)? Wait, no, the diagram: vertex \(B\) has angle \(28^\circ\), vertex \(A\) has angle \(83^\circ\), side \(a = 18\) is from \(B\) to \(C\), so side \(a\) is opposite angle \(A\) (angle at \(A\)). So angle \(A\) is \(83^\circ\), opposite side \(a = 18\). Angle \(B\) is \(28^\circ\), opposite side \(b\) (from \(A\) to \(C\)). So the Law of Sines is correct as \(\frac{\sin A}{a}=\frac{\sin B}{b}\). So the calculation is correct.

Wait, let's recalculate:

\(\sin 28^\circ \approx 0.4694715628\)

\(\sin 83^\circ \approx 0.9925461517\)

So \(b = \frac{18 \times 0.4694715628}{0.9925461517}\)

\(18 \times 0.4694715628 = 8.45048813\)

Divide by \(0.9925461517\): \(8.45048813 \div 0.9925461517 \approx 8.51\)

Answer:

\(8.51\)